The term "arity" is primarily used with reference to operations. If f is the functionf : Sn → S, where S is some set, then f is an operation and n is its arity.

Arities greater than 2 are seldom encountered in mathematics, except in specialized areas, and arities greater than 3 are seldom encountered in theoretical computer science (although in practical computer programming, it is common to define functions with more than 3 arguments). (In many computer programming languages, a "function" can also do other things in addition to returning a value, such as perform I/O to the console or a storage device, perform memory organization operations, etc., and so the mathematical "arity" of a program function is only defined by its number of calling arguments when the return value depends on all the arguments and only on them. Clouding the issue further is the fact that in many languages, a program function which evaluates a mathematical function may obtain mathematical arguments from other sources, such as global variables or shared memory, device input, or even the processor stack prior to the function's stack frame; whether these should be counted in the arity of the program function is not clear, and depends on one's perspective on defining the function. I.e., is the computer program "function" the code, or is it the mathematical component of the operation embodied in the code?)

Examples

The term "arity" is rarely employed in everyday usage. For example, rather than saying "the arity of the addition operation is 2" or "addition is an operation of arity 2" one usually says "addition is a binary operation".
In general, the naming of functions or operators with a given arity follows a convention similar to the one used for n-based numeral systems such as binary and hexadecimal. One combines a Latin prefix with the -ary ending; for example:

Nullary

Sometimes it is useful to consider a constant as an operation of arity 0, and hence call it nullary.

Also, in non-functional programming, a function without arguments can be meaningful and not necessarily constant (due to side effects). Often, such functions have in fact some hidden input which might be global variables, including the whole state of the system (time, free memory, ...) The latter are important examples which usually also exist in "purely" functional programming languages.

Binary

Most operators encountered in programming are of the binary form. For both programming and mathematics these can be the multiplication operator, the addition operator, the division operator. Logical predicates such as OR, XOR, AND, IMP are typically used as binary operators with two distinct operands.

Ternary

From C, C++, C#, Java, Perl and variants comes the ternary operator?:, which is a so-called conditional operator, taking three parameters.
Forth also contains a ternary operator, */, which multiplies the first two (one-cell) numbers, dividing by the third, with the intermediate result being a double cell number. This is used when the intermediate result would overflow a single cell.

n-ary

From a mathematical point of view, a function of n arguments can always be considered as a function of one single argument which is an element of some product space. However, it may be convenient for notation to consider n-ary functions, as for example multilinear maps (which are not linear maps on the product space, if n≠1).

The same is true for programming languages, where functions taking several arguments could always be defined as functions taking a single argument of some complex type such as a tuple, or in languages with higher-order functions, by currying.

Other names

Polyadic or multary (or multiary) means any number of operands (or parameters).

n-ary means n operands (or parameters), but is often used as a synonym of "polyadic".

An alternative nomenclature is derived in a similar fashion from the corresponding Greek roots; for example, medadic, monadic, dyadic, triadic, polyadic, and so on. Thence derive the alternative terms adicity and adinity for the Latin-derived arity.

These words are often used to describe anything related to that number (e.g., undenary chess is a chess variant with an 11×11 board, or the Millenary Petition of 1603).